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PHY 7 11 Classical Mechanics and Mathematical Methods 11-11:50 AM MWF Olin 107

PHY 7 11 Classical Mechanics and Mathematical Methods 11-11:50 AM MWF Olin 107 Plan for Lecture 16: Read Chapter 7 & Appendices A-D Generalization of the one dimensional wave equation  various mathematical problems and techniques including: Sturm- Liouville equations

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PHY 7 11 Classical Mechanics and Mathematical Methods 11-11:50 AM MWF Olin 107

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  1. PHY 711 Classical Mechanics and Mathematical Methods • 11-11:50 AM MWF Olin 107 • Plan for Lecture 16: • Read Chapter 7 & Appendices A-D • Generalization of the one dimensional wave equation  various mathematical problems and techniques including: • Sturm-Liouville equations • Orthogonal function expansions; Fourier analysis • Green’s functions methods • Laplace transformation • Contour integration methods PHY 711 Fall 2016 -- Lecture 16

  2. Sturm-Liouville Equations PHY 711 Fall 2016 -- Lecture 16

  3. PHY 711 Fall 2016 -- Lecture 16

  4. Linear second-order ordinary differential equations Sturm-Liouville equations applied force given functions solution to be determined Homogenous problem: F(x)=0 PHY 711 Fall 2016 -- Lecture 16

  5. Examples of Sturm-Liouville eigenvalue equations -- PHY 711 Fall 2016 -- Lecture 16

  6. Solution methods of Sturm-Liouville equations (assume all functions and constants are real): PHY 711 Fall 2016 -- Lecture 16

  7. Comment on “completeness” It can be shown that for any reasonable function h(x), defined within the interval a < x <b, we can expand that function as a linear combination of the eigenfunctionsfn(x) These ideas lead to the notion that the set of eigenfunctionsfn(x)form a ``complete'' set in the sense of ``spanning'' the space of all functions in the interval a < x <b, as summarized by the statement: PHY 711 Fall 2016 -- Lecture 16

  8. Variation approximation to lowest eigenvalue In general, there are several techniques to determine the eigenvalues ln and eigenfunctionsfn(x).When it is not possible to find the ``exact'' functions, there are several powerful approximation techniques. For example, the lowest eigenvalue can be approximated by minimizing the function where is a variable function which satisfies the correct boundary values. The ``proof'' of this inequality is based on the notion that can in principle be expanded in terms of the (unknown) exact eigenfunctionsfn(x): where the coefficients Cn can be assumed to be real. PHY 711 Fall 2016 -- Lecture 16

  9. Estimation of the lowest eigenvalue – continued: From the eigenfunctionequation, we know that It follows that: PHY 711 Fall 2016 -- Lecture 16

  10. Rayleigh-Ritz method of estimating the lowest eigenvalue PHY 711 Fall 2016 -- Lecture 16

  11. Green’s function solution methods Recall: PHY 711 Fall 2016 -- Lecture 16

  12. Solution to inhomogeneous problem by using Green’s functions Solution to homogeneous problem PHY 711 Fall 2016 -- Lecture 16

  13. Example Sturm-Liouville problem: PHY 711 Fall 2016 -- Lecture 16

  14. PHY 711 Fall 2016 -- Lecture 16

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  16. PHY 711 Fall 2016 -- Lecture 16

  17. PHY 711 Fall 2016 -- Lecture 16

  18. General method of constructing Green’s functions using homogeneous solution PHY 711 Fall 2016 -- Lecture 16

  19. PHY 711 Fall 2016 -- Lecture 16

  20. PHY 711 Fall 2016 -- Lecture 16

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